How to save each individual "object" in an obj file as separate obj files?
I created a quick scene with a few basic Meshes Joined together to form a single Mesh



In Edit mode I selected everything and pressed P to bring up the Separation menu and selected "By loose parts"



Now in Object mode you can see even the over lapping components are separate Meshes.



From the File->Export menu select Wavefront (.obj)


And check "Selection Only"



I did not look into it, but you may be able to use Python to select each Mesh in turn and call the Export module in a loop.